# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

October 2005 , Volume 13 , Issue 5

Special Issue
Recent Development on Differential Equations and Dynamical Systems: Part II

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2005, 13(5): 1097-1124 doi: 10.3934/dcds.2005.13.1097 +[Abstract](1046) +[PDF](378.4KB)
Abstract:
We will classify the path connected components of spaces of Sobolev maps between manifolds and study the strong and weak density of smooth maps in the spaces of Sobolev maps for the case the domain manifold has nonempty boundary and Dirichlet problems.
2005, 13(5): 1125-1152 doi: 10.3934/dcds.2005.13.1125 +[Abstract](1204) +[PDF](323.8KB)
Abstract:
We study non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomorphisms with unstable dimension 2 through a Hopf bifurcation. Using some recent abstract results about non-uniformly expanding maps with holes, by ourselves and by Dysman, we show that the Hausdorff dimension and the limit capacity (box dimension) of the repeller are strictly less than the dimension of the ambient manifold.
2005, 13(5): 1153-1186 doi: 10.3934/dcds.2005.13.1153 +[Abstract](1035) +[PDF](6002.3KB)
Abstract:
We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equation for the Euler equation. Based on the multiscale analysis in the Lagrangian formulation, we also derive the corresponding multiscale analysis in the Eulerian formulation. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress term, which provides a theoretical base for establishing systematic multiscale modeling of 2-D incompressible flow.
2005, 13(5): 1187-1201 doi: 10.3934/dcds.2005.13.1187 +[Abstract](991) +[PDF](292.5KB)
Abstract:
In the study of systems which combine slow and fast motions which depend on each other (fully coupled setup) whenever the averaging principle can be justified this usually can be done only in the sense of $L^1$-convergence on the space of initial conditions. When fast motions are hyperbolic (Axiom A) flows or diffeomorphisms (as well as expanding endomorphisms) for each freezed slow variable this form of the averaging principle was derived in [19] and [20] relying on some large deviations arguments which can be applied only in the Axiom A or uniformly expanding case. Here we give another proof which seems to work in a more general framework, in particular, when fast motions are some partially hyperbolic or some nonuniformly hyperbolic dynamical systems or nonuniformly expanding endomorphisms.
2005, 13(5): 1203-1215 doi: 10.3934/dcds.2005.13.1203 +[Abstract](1256) +[PDF](277.9KB)
Abstract:
I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for $\C^\infty$ Anosov diffeomorphisms, the dynamical determinant is an entire function.
2005, 13(5): 1217-1234 doi: 10.3934/dcds.2005.13.1217 +[Abstract](1044) +[PDF](248.5KB)
Abstract:
We prove the existence of reaction-diffusion traveling fronts in mean zero space-time periodic shear flows for nonnegative reactions including the classical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP nonlinearity, the minimal front speed is characterized by a variational principle involving the principal eigenvalue of a space-time periodic parabolic operator. Analysis of the variational principle shows that adding a mean-zero space time periodic shear flow to an existing mean zero space-periodic shear flow leads to speed enhancement. Computation of KPP minimal speeds is performed based on the variational principle and a spectrally accurate discretization of the principal eigenvalue problem. It shows that the enhancement is monotone decreasing in temporal shear frequency, and that the total enhancement from pure reaction-diffusion obeys quadratic and linear laws at small and large shear amplitudes.
2005, 13(5): 1235-1246 doi: 10.3934/dcds.2005.13.1235 +[Abstract](1264) +[PDF](242.9KB)
Abstract:
We study the propagation of a front arising as the asymptotic (macroscopic) limit of a model in spatial ecology in which the invasive species propagate by "jumps". The evolution of the order parameter marking the location of the colonized/uncolonized sites is governed by a (mesoscopic) integro-differential equation. This equation has structure similar to the classical Fisher or KPP - equation, i.e., it admits two equilibria, a stable one at $k$ and an unstable one at $0$ describing respectively the colonized and uncolonized sites. We prove that, after rescaling, the solution exhibits a sharp front separating the colonized and uncolonized regions, and we identify its (normal) velocity. In some special cases the front follows a geometric motion. We also consider the same problem in heterogeneous habitats and oscillating habitats. Our methods, which are based on the analysis of a Hamilton-Jacobi equation obtained after a change of variables, follow arguments which were already used in the study of the analogous phenomena for the Fisher/KPP - equation.
2005, 13(5): 1247-1256 doi: 10.3934/dcds.2005.13.1247 +[Abstract](1279) +[PDF](190.2KB)
Abstract:
We consider the partial analogue of the usual measurable Livsic theorem for Anosov diffeomorphims in the context of non-uniformly hyperbolic diffeomorphisms (Theorem 2). Our main application of this theorem is to the density of absolutely continuous measures (Theorem 1).
2005, 13(5): 1257-1276 doi: 10.3934/dcds.2005.13.1257 +[Abstract](1012) +[PDF](325.9KB)
Abstract:
The linearized Primitive Equations with vanishing viscosity are considered. Some new boundary conditions (of transparent type) are introduced in the context of a modal expansion of the solution which consist of an infinite sequence of integral equations. Applying the linear semi-group theory, existence and uniqueness of solutions is established. The case with nonhomogeneous boundary values, encountered in numerical simulations in limited domains, is also discussed.
2005, 13(5): 1277-1304 doi: 10.3934/dcds.2005.13.1277 +[Abstract](1263) +[PDF](358.6KB)
Abstract:
We consider the long time behavior of moments of solutions and of the solutions itself to dissipative Quasi-Geostrophic flow (QG) with sub-critical powers. The flow under consideration is described by the nonlinear scalar equation

$\frac{\partial \theta}{\partial t} + u\cdot \nabla \theta + \kappa (-\Delta)^{\alpha}\theta =f$, $\theta|_{t=0}=\theta_0$

Rates of decay are obtained for moments of the solutions, and lower bounds of decay rates of the solutions are established.

2005, 13(5): 1305-1324 doi: 10.3934/dcds.2005.13.1305 +[Abstract](1293) +[PDF](302.5KB)
Abstract:
In this paper we develop the theory of polymorphisms of measure spaces, which is a generalization of the theory of measure-preserving transformations. We describe the main notions and discuss relations to the theory of Markov processes, operator theory, ergodic theory, etc. We formulate the important notion of quasi-similarity and consider quasi-similarity between polymorphisms and automorphisms.

The question is as follows: is it possible to have a quasi-similarity between a measure-preserving automorphism $T$ and a polymorphism $\Pi$ (that is not an automorphism)? In less definite terms: what kind of equivalence can exist between deterministic and random (Markov) dynamical systems? We give the answer: every nonmixing prime polymorphism is quasi-similar to an automorphism with positive entropy, and every $K$-automorphism $T$ is quasi-similar to a polymorphism $\Pi$ that is a special random perturbation of the automorphism $T$.
2005, 13(5): 1325-1342 doi: 10.3934/dcds.2005.13.1325 +[Abstract](1140) +[PDF](269.2KB)
Abstract:
It is shown that periodic solutions of a delay differential equation approach a square wave if a parameter becomes large. The equation models short-term prize fluctuations. The proof relies on the fact that the branches of the unstable manifold at equilibrium tend to the periodic orbit.

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